How to loop cut a cylinder into squares
This can be done by starting from a circle instead of a cylinder.

Open the properties panel (P) in the 3d view and enable 'Length' in the 'Mesh Display' section to view edge lengths in the 3D view:



In front view, with all edges of the circle selected, extrude with E and constrain to the Z axis by pressing Z.

You can now extrude upwards and match the extrusion length with the original edge length. In this case it is 0.196. To do this perfectly you can just type whatever the edge length is (0.196) while extruding and constraining:



The last step is to add an array modifier to to create the other 'rows' of the cylinder. As long as the object has it's rotation and scale applied you can just set an offset of '1' in the Z offset field: